In this course we will use Sage computer algebra system (CAS), which is a free software.
The Sage projects are created to help you learn new concepts. Sage is very useful in visualizing graphs
and surfaces in three dimensions. Matlab (commercial software) is also available at engineeering labs. Its free version
is called Octave. The university has a license for computer algebra system Mathematica, so it is free to use for its students. A student can also use free CASs: SymPy (based on Python), or Maxima.

Section 14.6. The Chain Rule

which expresses the change in entropy \( {\text d}S \) of a system as a function of the intensive quantities
temperature T, presure p, and ith chemical potential \( \mu_i \) and
of the differentials of the intensive quantities energy U, volume V, and ith particle number
\( N_i . \) The equation may be seen as a particular case of the chain rule. In other words

These equations are known as "equations of state" with respect to the internal energy. (Note - the relation between
pressure, volume, temperature, and particle number which is commonly called "the equation of state" is just one of
many possible equations of state.) If we know all s+2 of the above equations of state, we may reconstitute the
fundamental equation and recover all thermodynamic properties of the system.

The fundamental equation can be solved for any other differential and similar expressions can be found. For example,
we may solve for \( {\text d}S \) and find that